Express \(\cos 3\theta\) in terms of \(\cos\theta\). Show that, for any real \(\theta\), \[ \cos\theta - \tfrac{1}{2}(1-\cos 2\theta) < \cos 3\theta < \cos 2\theta + \tfrac{1}{4}(1-\cos\theta). \]
Prove the formula \[ \frac{1}{(x^2+1)^n} = \frac{1}{2n-2}\frac{d}{dx}\left(\frac{x}{(x^2+1)^{n-1}}\right) + \frac{2n-3}{2n-2}\frac{1}{(x^2+1)^{n-1}} \] for \(n \ge 2\), and hence obtain a recurrence relation for the indefinite integral \[ I_n = \int \frac{dt}{(1+t^2)^n}. \] Evaluate \[ \int_0^1 \frac{dt}{(1+t^2)^3}. \]
Given that \(f_0(x)>0\) for \(x \ge 0\), and that \[ f_n(x) = \int_0^x f_{n-1}(t)dt \quad (n=1,2,3,\dots), \] prove that \[ \frac{f_n(x)}{x^n} > \frac{f_{n+1}(x)}{x^{n+1}} \] for \(n \ge 1\) and \(x > 0\). By repeated integration by parts, verify the formula \[ f_n(x) = \frac{1}{(n-1)!} \int_0^x (x-u)^{n-1} f_0(u)du \] for \(n \ge 1\) and \(x \ge 0\).
A curve is given parametrically by the equations \[ x = a\cos^3 t \quad y = a\sin^3 t. \] Find the parametric equations of the locus of its centre of curvature.
Sketch the curve \[ x=t^2+1 \quad y=t(t^2-4). \] Show that it has a loop, and find the area of this loop.
(i) Given that \(\alpha\) and \(\beta\) are the roots of \[ x^2 - px + q = 0, \] form the equation whose roots are \(\alpha^3 - \frac{1}{\beta^3}\), \(\beta^3 - \frac{1}{\alpha^3}\). (ii) Given that the equation \[ x^n - ax^2 + bx - c = 0, \quad (c \neq 0, n > 2), \] has a thrice repeated root \(\xi\), establish the relations \[ \xi = \frac{(n-1)b}{2(n-2)a} = \frac{2nc}{(n-1)b}, \] and \[ \xi^n = \frac{2c}{(n-1)(n-2)}. \]
Find the sum of the series \[ x+2^2x^2+3^2x^3+4^2x^4+\dots+n^2x^n. \] Hence, or otherwise, evaluate \[ S_n = 1-2^2+3^2-\dots+(-1)^{n-1}n^2. \]
Given that \(a_r, b_r\) and \(c_r\) are all real and positive numbers for \(r=1, 2, \dots, n\), and that \[ a_r^2 = b_r^2+c_r^2, \quad r=1, 2, \dots, n, \] \[ A_n = \sum_{r=1}^n a_r, \quad B_n = \sum_{r=1}^n b_r, \quad C_n = \sum_{r=1}^n c_r, \] prove by induction, or otherwise, that for \(n \ge 1\), \[ A_n^2 \ge B_n^2 + C_n^2. \]
Solve completely the equation \[ \sin 3x = \cos 4x. \] Hence, or otherwise, find all the roots of \[ 8y^4+4y^3-8y^2-3y+1=0. \]
A tetrahedron \(ABCD\) has edges of lengths \(AB=AC=AD=a\), and \(BC=CD=DB=b\). A sphere is inscribed in this tetrahedron so that it touches the four faces. Find the radius of this sphere in terms of \(a\) and \(b\).